arxiv: 2604.27235 · v1 · submitted 2026-04-29 · 🧮 math.RT · math.AT· math.GR· math.NT

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Asymptotic Vanishing of Stiefel--Whitney Classes for GL_n(mathbb{F}_q)

Anwesh Ray

Pith reviewed 2026-05-07 10:30 UTC · model grok-4.3

classification 🧮 math.RT math.ATmath.GRmath.NT

keywords Stiefel-Whitney classesorthogonal representationsgeneral linear groupsfinite fieldsasymptotic vanishingspinorial representationscharacter values2-adic divisibility

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The pith

For fixed odd q, as n grows the proportion of irreducible orthogonal representations of GL_n(F_q) with trivial first and second Stiefel-Whitney classes tends to 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper connects Stiefel-Whitney classes of orthogonal representations of finite general linear groups to the 2-adic divisibility of their character values at involutions. Using this link, it proves that when q is a fixed odd prime power and n tends to infinity, those character values become divisible by arbitrarily high powers of 2 for almost every irreducible orthogonal representation. This forces the first and second Stiefel-Whitney classes to vanish on a set of representations whose proportion approaches 1. When q is congruent to 1 modulo 4 the fourth class also vanishes asymptotically. The result implies that almost every orthogonal representation becomes spinorial in the large-rank regime.

Core claim

For fixed odd q we show that as n tends to infinity the values of irreducible orthogonal characters of GL_n(F_q) become highly divisible by powers of 2 for almost all representations. Consequently the proportion of irreducible orthogonal representations with trivial first and second Stiefel-Whitney classes tends to 1, and if q ≡ 1 mod 4 the same holds for the fourth Stiefel-Whitney class. In particular almost all orthogonal representations are spinorial in the large-rank limit. When the rank is fixed and q tends to infinity the behavior differs; for GL_2(F_q) the second Stiefel-Whitney class vanishes with limiting probability 5/16.

What carries the argument

Recent formulas that express the Stiefel-Whitney classes of orthogonal representations directly in terms of the values of the character at elements of order dividing 2, together with an asymptotic analysis of the 2-adic valuations of those character values as the rank n increases.

If this is right

Almost all orthogonal representations of GL_n(F_q) become spinorial when n is large.
The second Stiefel-Whitney class vanishes for a proportion of representations that approaches 1.
When q ≡ 1 mod 4 the fourth Stiefel-Whitney class likewise vanishes asymptotically.
The vanishing probabilities are different when n is held fixed and q grows; for example the second class vanishes with probability 5/16 among orthogonal irreps of GL_2(F_q).
The asymptotic behavior depends on whether the rank or the field size is sent to infinity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result points to a form of stability for characteristic classes of orthogonal representations once the rank is allowed to grow.
It may be possible to lift the argument to other families of finite groups of Lie type where similar character-value formulas exist.
Numerical checks for moderate n could confirm how quickly the proportion of vanishing classes approaches 1.
The contrast between the two growth regimes suggests that representation-theoretic limits are sensitive to the order in which parameters tend to infinity.

Load-bearing premise

The formulas expressing Stiefel-Whitney classes in terms of character values at elements of order dividing 2 are valid and apply to the irreducible orthogonal representations of GL_n(F_q).

What would settle it

An explicit sequence of n_k tending to infinity together with a positive-density subset of irreducible orthogonal representations for each n_k whose second Stiefel-Whitney class remains nontrivial would falsify the asymptotic vanishing claim.

read the original abstract

We study the asymptotic behavior of Stiefel--Whitney classes of irreducible orthogonal representations of the finite general linear groups $\mathrm{GL}_n(\mathbb{F}_q)$. Building on recent formulas expressing these classes in terms of character values at elements of order dividing $2$, we relate questions about characteristic classes to problems of $2$-adic divisibility of character values. For fixed odd $q$, we show that as $n \to \infty$, the values of irreducible orthogonal characters become highly divisible by powers of $2$ for almost all representations. As a consequence, the proportion of irreducible orthogonal representations with trivial first and second Stiefel--Whitney classes tends to $1$, and if $q \equiv 1 \pmod{4}$, the same holds for the fourth Stiefel--Whitney class. In particular, almost all orthogonal representations are spinorial in the large rank limit. In contrast, when the rank is fixed and $q \to \infty$, the behavior is markedly different. Focusing on $\mathrm{GL}_2(\mathbb{F}_q)$, we show that the second Stiefel--Whitney class vanishes with limiting probability $5/16$ among irreducible orthogonal representations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows that for fixed odd q, almost all irreducible orthogonal representations of GL_n(F_q) have trivial first and second Stiefel-Whitney classes as n goes to infinity, with the fourth also vanishing when q ≡ 1 mod 4, and gives a precise 5/16 limiting probability for the second class in the GL_2 case.

read the letter

The main result is that the first and second Stiefel-Whitney classes vanish for a proportion tending to 1 of the irreducible orthogonal representations of GL_n(F_q) when n is large and q is fixed and odd. When q is 1 mod 4 the fourth class does the same. For GL_2 the second class vanishes with limiting probability exactly 5/16 as q grows. This is the concrete new content: the limiting proportions and the explicit small-rank calculation, obtained by reducing the topological question to 2-adic divisibility of character values at 2-elements via the cited formulas, then applying standard Deligne-Lusztig estimates to show the required divisibility holds for all but an o(1) fraction of the representations. The contrast between the large-n regime (vanishing) and the large-q regime (non-vanishing with positive probability) is also useful to record. The reduction itself is straightforward once the formulas are granted, and the argument structure is internally consistent with no obvious uniformity problems in the orthogonal subset. The main soft spot is the dependence on the prior formulas that express the classes in terms of those character values; if those formulas have any edge cases or restrictions when applied to orthogonal representations, the count of representations satisfying the divisibility condition could shift. The asymptotic control also needs the exceptional set to be shown o(1) without hidden constants depending on n, but the stress-test outline suggests the estimates are the usual ones and do not break. No circularity or invented parameters appear. This is for specialists in the representation theory of finite groups of Lie type who care about links to topology or about character divisibility by primes. A reader already following work on Stiefel-Whitney classes or 2-adic properties of characters will find the limiting statements and the GL_2 probability worth checking. The paper is coherent on its own terms and the claims are specific enough to be refereeable, so it deserves a serious review rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper studies the asymptotic behavior of Stiefel-Whitney classes of irreducible orthogonal representations of GL_n(F_q) for odd q. Building on formulas expressing these classes via character values at 2-elements, it shows that for fixed odd q, as n→∞ the character values at such elements become highly 2-divisible for almost all irreducible orthogonal representations. Consequently, the proportion with trivial first and second Stiefel-Whitney classes tends to 1 (and likewise for the fourth class when q≡1 mod 4), so almost all such representations are spinorial in the large-rank limit. In contrast, for fixed rank GL_2(F_q) as q→∞, the second Stiefel-Whitney class vanishes with limiting probability 5/16 among irreducible orthogonal representations.

Significance. If the claims hold, the work establishes a precise asymptotic vanishing result for characteristic classes in the representation theory of finite groups of Lie type, linking it directly to 2-adic divisibility properties of character values at semisimple elements. The explicit probability computation in the GL_2 case provides a concrete, falsifiable prediction that contrasts the large-n and large-q regimes. The reduction via existing formulas for the classes, combined with standard Deligne-Lusztig estimates, supplies a clean and reproducible argument structure without free parameters or ad-hoc fitting.

minor comments (3)

The abstract and introduction refer to 'recent formulas' for expressing Stiefel-Whitney classes in terms of character values; these should be cited with explicit reference numbers and a brief statement of the precise hypotheses under which they apply to orthogonal representations.
In the GL_2(F_q) analysis, the limiting probability 5/16 is stated as a consequence of counting orthogonal representations and their character values; a short table or explicit enumeration of the contributing Lusztig series (or conjugacy classes of 2-elements) would make the arithmetic transparent.
Notation for the orthogonal subset of irreducible representations and the precise meaning of 'highly divisible by powers of 2' (e.g., valuation bounds uniform in n) should be fixed early and used consistently in the asymptotic statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, which correctly captures the main results on the asymptotic vanishing of Stiefel-Whitney classes for irreducible orthogonal representations of GL_n(F_q) as n tends to infinity with q fixed and odd. We appreciate the recommendation for minor revision and the recognition of the link to 2-adic divisibility of character values. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation reduces SW vanishing to independent asymptotic divisibility estimates

full rationale

The paper invokes recent formulas to express Stiefel-Whitney classes via character values at 2-elements, then proves that for fixed odd q the proportion of irreducible orthogonal representations whose character values at those elements are highly 2-divisible tends to 1 as n→∞, using Deligne-Lusztig character estimates and counting arguments on Lusztig series. This divisibility limit is established by direct asymptotic analysis of representation counts and character bounds that do not depend on re-deriving or fitting the input formulas; the formulas serve only as a reduction step whose validity is external to the limit computation. No step equates a prediction to its own fitted input, renames a known result, or relies on a self-citation chain that itself assumes the target vanishing. The fixed-n, q→∞ contrast further confirms the asymptotic claim is not tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on prior formulas linking Stiefel-Whitney classes to character values at 2-elements and on standard asymptotic counting arguments in representation theory; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption Formulas expressing Stiefel-Whitney classes of orthogonal representations in terms of character values at elements of order dividing 2
The paper explicitly builds on these recent formulas as the starting point for the divisibility analysis.
standard math Standard facts from the character theory of finite groups of Lie type and algebraic topology
Background results on irreducible characters and Stiefel-Whitney classes are invoked without proof.

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discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages

[1]

Translated from the second French edition by Leonard L. Scott. [Spr70] T. A. Springer. Characters of special groups. InSeminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), Lecture Notes in Math., Vol. 131, pages 121–166. Springer, Berlin-New York,

1968
[2]

The divisibility ofGL(n, q)character values.arXiv preprint arXiv:2408.14046,

[SS24] Varun Shah and Steven Spallone. The divisibility ofGL(n, q)character values.arXiv preprint arXiv:2408.14046,

work page arXiv